On an extremal problem for harmonic maps conformal at a point
Abstract
Let 𝔻 denote the unit disc in ℂ. For a domain D⊂ℂ and a point p∈ D, let MD(p) denote the supremum of |df0| over all harmonic maps f:𝔻→ D with f(0)=p whose differential df0 at 0∈ 𝔻 is conformal. If f:𝔻→ D is a conformal diffeomorphism onto D with f(0)=p, then |df0|≤ MD(p). In a recent paper, the authors proved that equality holds when D=𝔻, and they asked whether equality can hold only when D is a round disc. We give a negative answer by proving that, among bounded convex pointed domains p∈ D⊂ℂ and up to translations, rotations, and reflections, equality holds if and only if, after moving p to the origin, D=F(𝔻) where F:𝔻→ℂ is a holomorphic map with F(0)=0 and F'(z)=c/(1+az+λz2), where c>0, |λ|<1, and |a−a̅λ|<1−|λ|2. This family contains strongly convex examples which are not round discs.
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BibTeX
@misc{forstneric2026extremal,
title = {{On an extremal problem for harmonic maps conformal at a point}},
author = {Franc Forstneric and David Kalaj},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.09576},
eprint = {2605.09576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.09576},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On an extremal problem for harmonic maps conformal at a point
AU - Franc Forstneric
AU - David Kalaj
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2605.09576
UR - https://arxiv.org/abs/2605.09576
AB - Let 𝔻 denote the unit disc in ℂ. For a domain D⊂ℂ and a point p∈ D, let M_D(p) denote the supremum of |df_0| over all harmonic maps f:𝔻→ D with f(0)=p whose differential df_0 at 0∈ 𝔻 is conformal. If f:𝔻→ D is a conformal diffeomorphism onto D with f(0)=p, then |df_0|≤ M_D(p). In a recent paper, the authors proved that equality holds when D=𝔻, and they asked whether equality can hold only when D is a round disc. We give a negative answer by proving that, among bounded convex pointed domains p∈ D⊂ℂ and up to translations, rotations, and reflections, equality holds if and only if, after moving p to the origin, D=F(𝔻) where F:𝔻→ℂ is a holomorphic map with F(0)=0 and F'(z)=c/(1+az+λz^2), where c>0, |λ|<1, and |a−a̅λ|<1−|λ|^2. This family contains strongly convex examples which are not round discs.
ER -